On Static n-body Configurations in Relativity

TL;DR

This paper investigates static n-body configurations in General Relativity, proving their non-existence under certain symmetry and separation conditions, thereby advancing understanding of gravitational equilibrium states.

Contribution

The paper provides a solution to the static n-body problem in cases with reflection symmetry across a noncompact surface, extending previous non-existence results.

Findings

No static n-body configurations exist under the specified conditions.

The result applies when a reflection symmetry across a noncompact surface is present.

The work broadens the understanding of gravitational equilibrium in relativistic settings.

Abstract

The static n-body problem of General Relativity states that there are, under a reasonable energy condition, no static $n$-body configurations for $n > 1$, provided the configuration of the bodies satisfies a suitable separation condition. In this paper we solve this problem in the case that there exists a closed, noncompact, totally geodesic surface disjoint from the bodies. This covers the situation where the configuration has a reflection symmetry across a noncompact surface disjoint from the bodies.